Fukaya-category embedding conjecture for the open Calabi–Yau threefold $Y_\Phi$

Let $S$ be a complex curve with marked-point divisor $D$, let $m\geq 1$, let $\Phi$ be an admissible generic point of the Hitchin base, and let $Y_\Phi$ be the associated open Calabi–Yau threefold. Let $Q_{\mathcal{T},m}$ be the quiver associated to a triangulation $\mathcal{T}$, let $W$ be a potential on it, and let $\mathcal{D}(Q_{\mathcal{T},m},W)$ denote the corresponding $3$-dimensional Calabi–Yau category. For every admissible Kähler-form class $[\omega]\in H^2(Y_\Phi,\mathbb{R})$, there should exist a potential $W$, defined up to right-equivalence, and a fully faithful embedding

$$\mathcal{D}(Q_{\mathcal{T},m},W)\hookrightarrow \mathcal{DF}(Y_\Phi,b_0)$$

for a suitable background class $b_0\in H^2(Y_\Phi,\mathbb{Z})$. This conjecturally realizes the quiver-with-potential category inside the Fukaya category of $Y_\Phi$; the existence of the embedding and the required potential remain open.